Question 1

Solve the differential equation: y' = $e ^ { y }$ sin t.

Accepted Answer

A)  y = cos(ln t) + C 
B)  y = $\mathrm { e } ^ { - \cos }$ + C 
C)  y = ln(-sin t) + C 
D)  y = -ln(cos t + C 
A)  y = cos(ln t) + C 
B)  y = $\mathrm { e } ^ { - \cos }$ + C 
C)  y = ln(-sin t) + C 
D)  y = -ln(cos t + C

Question 2

Consider the differential equation   =   - 4y + 3. Which of the following could be a graph of solutions to this differential equation?

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 3

A sports enthusiast drinks 2 liters of water per hour. Water is eliminated from the body at a rate proportional to the amount of water in the body (due to perspiration). Write a differential equation satisfied by f(t), the amount of water in the body. Does this equation accurately describe this situation: $\mathrm { y } ^ { \prime } = 2 - \mathrm { ky } ?$

Accepted Answer

A) True 
 B)False

Question 4

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false:  
-$f ^ { \prime }$ (0) = 2

Accepted Answer

A) True 
 B)False

Question 5

A cool object is to be heated to a maximum temperature M = M°C. At any time t, the rate at which the temperature rises is proportional to the difference between the actual temperature and the maximal temperature. If the object is originally 0°C, find and solve a differential equation describing this situation. Is this the solution: $y ( t ) = M - M e ^ { - k t }$

Accepted Answer

A) True 
 B)False

Question 6

Suppose the graph below gives a solution to the differential equation $\frac { \mathrm { dP } } { \mathrm { ds } }$ = g(P) where P is the price of a product and s is the weekly sales.   Which of the following statements is/are true? 
(I) g(M) = 0
(II) $g ^ { \prime } ( M ) = 0$
(III) $g \left( \frac { \mathrm { M } } { 2 } \right) _ { = 0 }$
(IV) g( $\mathrm { P } _ { 0 }$ ) > 0

Accepted Answer

A)  I only 
B)  IV only 
C)  I, III, and IV 
D)  I and IV 
E)  I, II, and IV 
A)  I only 
B)  IV only 
C)  I, III, and IV 
D)  I and IV 
E)  I, II, and IV

Question 7

A nutritionist proposes the following model for weight loss on a program she is developing: $\frac { \mathrm { dw } } { \mathrm { dt } }$ + 0.006w = $\frac { 1 } { 3600 }$ C where w(t) is a person's weight (in pounds) after t days of consuming exactly C calories per day. A person weighing 180 pounds goes on this diet program consuming 2400 calories per day. Use the above model to predict how long will it take this person to lose 15 pounds.

Accepted Answer

A)  41 days 
B)  37 days 
C)  39 days 
D)  35 days 
A)  41 days 
B)  37 days 
C)  39 days 
D)  35 days

Question 8

Solve the differential equation with the given initial condition. 
-$y ^ { \prime } = y ^ { 2 } , y ( 0 ) = 1$

Accepted Answer

A)  t + 1 
B)  $\frac { 1 } { t + 1 }$ 
C)  t - 1 
D)  $\frac { 1 } { 1 - t }$ 
A)  t + 1 
B)  $\frac { 1 } { t + 1 }$ 
C)  t - 1 
D)  $\frac { 1 } { 1 - t }$

Question 9

A jug of milk at 50° is placed outdoors at a temperature of 100°. If after 5 minutes the temperature of the milk is 60°, write the equation giving the temperature of the milk as a function of time. Enter your answer exactly as:  
T = a $\mathrm { e } ^ { \mathrm { b } }$ + c

Accepted Answer

The answer of A jug of milk at 50° is...

Question 10

Solve the problem.
-An initial deposit of $24,000 is made into an account that earns 5% compounded continuously. Money is then withdrawn at a constant rate of $4000 a year until the amount in the account is 0. Find the equation for the amount in the account at any time t. When is the amount 0?

Accepted Answer

A)  A = 60,000 - 36,000 $e ^ { 0.05 t }$ = 10.017 years 
B)  A = 80,000 - 56,000 $\mathrm { e } ^ { 0.05 \mathrm { t } }$= 8.352 years 
C)  A = 80,000 - 56,000 $e ^ { 0.05 t }$= 7.134 years 
D)  A = 60,000 - 36,000 $e ^ { 0.05 t }$= 8.352 years 
A)  A = 60,000 - 36,000 $e ^ { 0.05 t }$ = 10.017 years 
B)  A = 80,000 - 56,000 $\mathrm { e } ^ { 0.05 \mathrm { t } }$= 8.352 years 
C)  A = 80,000 - 56,000 $e ^ { 0.05 t }$= 7.134 years 
D)  A = 60,000 - 36,000 $e ^ { 0.05 t }$= 8.352 years

Question 11

Suppose that y = f(t) satisfies the differential equation $y ^ { \prime }$ = $t ^ { 2 }$ + 3y, f(0) = 2. If Euler's method with $\mathrm { n } = 10$ is used to construct an approximation p(t) to f(t) for 0 ≤ t ≤ 1, find p(0.1).

Accepted Answer

A)  2.2 
B)  2.6 
C)  3.1 
D)  6.0 
E)  none of these 
A)  2.2 
B)  2.6 
C)  3.1 
D)  6.0 
E)  none of these

Question 12

Given the differential equation: $y ^ { \prime } = 3 t ^ { 2 } ( y - 7 )$ , is this the solution $y = 7 + \mathrm { ce } ^ { t ^ { 3 } } ?$

Accepted Answer

A) True 
 B)False

Question 13

One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. $y ^ { \prime } = \cos y ; y ( 0 ) = - \frac { \pi } { 4 } ; y ( 0 ) = \frac { 5 \pi } { 4 }$ Do these graphs represent the situation?

Accepted Answer

A) True 
 B)False

Question 14

Given the differential equation with the given initial condition: $\frac { d y } { d t } = 3 t ^ { 2 } + \sin t ; y ( 0 ) = 2$ is this the solution $y = t ^ { 3 } - \cos t + 1 ?$

Accepted Answer

A) True 
 B)False

Question 15

Suppose the relationship between the price p, of a product and the weekly sales, s, of the product is given by the differential equation $\frac { d p } { d s } = - \frac { 1 } { 10 } ( s + 3 )$ Then

Accepted Answer

A)  as sales increase, the price increases. 
B)  as the price increases the rate of change of the price also increases. 
C)  the rate of the decrease of the price is proportional to the sales. 
D)  s = 0 is a constant solution to this differential equation. 
E)  all of these 
A)  as sales increase, the price increases. 
B)  as the price increases the rate of change of the price also increases. 
C)  the rate of the decrease of the price is proportional to the sales. 
D)  s = 0 is a constant solution to this differential equation. 
E)  all of these

Question 16

Given the differential equation with the given initial condition: $y ^ { \prime } = \sqrt { \frac { t + 1 } { y } } ; y ( 0 ) = 4$ is this the solution $y = \left( ( t + 1 ) ^ { 3 / 2 } + 4 \right) ^ { 2 / 3 } \text { ? }$

Accepted Answer

A) True 
 B)False

Question 17

Find the constant solutions to the differential equation: $y ^ { \prime } = y ^ { 2 } e ^ { t } - 2 y e ^ { t }$ .
Enter just one integer or two separated by a comma (no label).

Accepted Answer

The answer of Find the constant solutions to the differential...

Question 18

Given $y ^ { \prime }$ = $e ^ { y }$ - 1. On a ty-coordinate system sketch the solutions corresponding to the initial conditions $y ( 0 ) = - 1$ and $y ( 0 ) = 1$ . Does this graph represent the situation?

Accepted Answer

A) True 
 B)False

Question 19

Solve the differential equation with the given initial condition. 
-$y ^ { \prime } = \sin t \cos ^ { 3 } t , y \left( \frac { \pi } { 3 } \right) = 0$

Accepted Answer

A)  y = $\frac { 1 } { 64 }$ - $\frac { \cos ^ { 4 } t } { 4 }$ 
B)  y = $\frac { \cos ^ { 4 } t } { 4 }$ 
C)  y = 16 + $\frac { \cos ^ { 4 } t } { 4 }$ 
D)  y = 64 - $\cos 4$ t 
A)  y = $\frac { 1 } { 64 }$ - $\frac { \cos ^ { 4 } t } { 4 }$ 
B)  y = $\frac { \cos ^ { 4 } t } { 4 }$ 
C)  y = 16 + $\frac { \cos ^ { 4 } t } { 4 }$ 
D)  y = 64 - $\cos 4$ t

Question 20

Use Euler's method with n = 5 on the interval 0 ≤ t ≤ $\frac { 1 } { 2 }$ to approximate the solution f(t) to $y ^ { \prime } = 3 y , y ( 0 ) = 1$ Is the following the correct answer? $\begin{array} { l } 
\mathrm { t } _ { 0 } = 0 ; \quad \mathrm { y } _ { 0 } = 1 \
\mathrm { t } _ { 1 } = 0.1 ; \quad \mathrm { y } _ { 1 } = 1.3 \
\mathrm { t } _ { 2 } = 0.2 ; \quad \mathrm { y } _ { 2 } = 1.69 \
\mathrm { t } 3 = 0.3 ; \quad \mathrm { y } _ { 3 } = 2.197 \
\mathrm { t } _ { 4 } = 0.4 ; \quad \mathrm { y } _ { 4 } = 2.8561 \
5 = 0.5 ; \quad \text { y5 } = 3.71293 \
\end{array}$

Accepted Answer

A) True 
 B)False

Solve the differential equation: y' = $e ^ { y }$ sin t.

Consider the differential equation = - 4y + 3. Which of the following could be a graph of solutions to this differential equation?

Solve the differential equation with the given initial condition. - $y ^ { \prime } = y ^ { 2 } , y ( 0 ) = 1$

A jug of milk at 50° is placed outdoors at a temperature of 100°. If after 5 minutes the temperature of the milk is 60°, write the equation giving the temperature of the milk as a function of time. Enter your answer exactly as: T = a $\mathrm { e } ^ { \mathrm { b } }$ + c

Solve the problem. -An initial deposit of $24,000 is made into an account that earns 5% compounded continuously. Money is then withdrawn at a constant rate of $4000 a year until the amount in the account is 0. Find the equation for the amount in the account at any time t. When is the amount 0?

Suppose that y = f(t) satisfies the differential equation $y ^ { \prime }$ = $t ^ { 2 }$ + 3y, f(0) = 2. If Euler's method with $\mathrm { n } = 10$ is used to construct an approximation p(t) to f(t) for 0 ≤ t ≤ 1, find p(0.1).

Given the differential equation: $y ^ { \prime } = 3 t ^ { 2 } ( y - 7 )$ , is this the solution $y = 7 + \mathrm { ce } ^ { t ^ { 3 } } ?$

Given the differential equation with the given initial condition: $\frac { d y } { d t } = 3 t ^ { 2 } + \sin t ; y ( 0 ) = 2$ is this the solution $y = t ^ { 3 } - \cos t + 1 ?$

Suppose the relationship between the price p, of a product and the weekly sales, s, of the product is given by the differential equation $\frac { d p } { d s } = - \frac { 1 } { 10 } ( s + 3 )$ Then

Given the differential equation with the given initial condition: $y ^ { \prime } = \sqrt { \frac { t + 1 } { y } } ; y ( 0 ) = 4$ is this the solution $y = \left( ( t + 1 ) ^ { 3 / 2 } + 4 \right) ^ { 2 / 3 } \text { ? }$

Find the constant solutions to the differential equation: $y ^ { \prime } = y ^ { 2 } e ^ { t } - 2 y e ^ { t }$ . Enter just one integer or two separated by a comma (no label).

Solve the differential equation with the given initial condition. - $y ^ { \prime } = \sin t \cos ^ { 3 } t , y \left( \frac { \pi } { 3 } \right) = 0$

Use Euler's method with n = 5 on the interval 0 ≤ t ≤ $\frac { 1 } { 2 }$ to approximate the solution f(t) to $y ^ { \prime } = 3 y , y ( 0 ) = 1$ Is the following the correct answer? =0;=1 =0.1;=1.3 =0.2;=1.69 3=0.3;=2.197 =0.4;=2.8561 5=0.5; y5 =3.71293

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 10: Differential Equations

Solve the differential equation: y' = eye ^ { y }ey sin t.

Consider the differential equation = - 4y + 3. Which of the following could be a graph of solutions to this differential equation?

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false: - f′f ^ { \prime }f′ (0) = 2

Solve the differential equation with the given initial condition. - y′=y2,y(0)=1y ^ { \prime } = y ^ { 2 } , y ( 0 ) = 1y′=y2,y(0)=1

A jug of milk at 50° is placed outdoors at a temperature of 100°. If after 5 minutes the temperature of the milk is 60°, write the equation giving the temperature of the milk as a function of time. Enter your answer exactly as: T = a eb\mathrm { e } ^ { \mathrm { b } }eb + c

Solve the problem. -An initial deposit of $24,000 is made into an account that earns 5% compounded continuously. Money is then withdrawn at a constant rate of $4000 a year until the amount in the account is 0. Find the equation for the amount in the account at any time t. When is the amount 0?

Suppose that y = f(t) satisfies the differential equation y′y ^ { \prime }y′ = t2t ^ { 2 }t2 + 3y, f(0) = 2. If Euler's method with n=10\mathrm { n } = 10n=10 is used to construct an approximation p(t) to f(t) for 0 ≤ t ≤ 1, find p(0.1).

Given the differential equation: y′=3t2(y−7)y ^ { \prime } = 3 t ^ { 2 } ( y - 7 )y′=3t2(y−7) , is this the solution y=7+cet3?y = 7 + \mathrm { ce } ^ { t ^ { 3 } } ?y=7+cet3?

Given the differential equation with the given initial condition: dydt=3t2+sin⁡t;y(0)=2\frac { d y } { d t } = 3 t ^ { 2 } + \sin t ; y ( 0 ) = 2dtdy​=3t2+sint;y(0)=2 is this the solution y=t3−cos⁡t+1?y = t ^ { 3 } - \cos t + 1 ?y=t3−cost+1?

Suppose the relationship between the price p, of a product and the weekly sales, s, of the product is given by the differential equation dpds=−110(s+3)\frac { d p } { d s } = - \frac { 1 } { 10 } ( s + 3 )dsdp​=−101​(s+3) Then

Given the differential equation with the given initial condition: y′=t+1y;y(0)=4y ^ { \prime } = \sqrt { \frac { t + 1 } { y } } ; y ( 0 ) = 4y′=yt+1​​;y(0)=4 is this the solution y=((t+1)3/2+4)2/3 ? y = \left( ( t + 1 ) ^ { 3 / 2 } + 4 \right) ^ { 2 / 3 } \text { ? }y=((t+1)3/2+4)2/3 ?

Find the constant solutions to the differential equation: y′=y2et−2yety ^ { \prime } = y ^ { 2 } e ^ { t } - 2 y e ^ { t }y′=y2et−2yet . Enter just one integer or two separated by a comma (no label).

Given y′y ^ { \prime }y′ = eye ^ { y }ey - 1. On a ty-coordinate system sketch the solutions corresponding to the initial conditions y(0)=−1y ( 0 ) = - 1y(0)=−1 and y(0)=1y ( 0 ) = 1y(0)=1 . Does this graph represent the situation?

Solve the differential equation with the given initial condition. - y′=sin⁡tcos⁡3t,y(π3)=0y ^ { \prime } = \sin t \cos ^ { 3 } t , y \left( \frac { \pi } { 3 } \right) = 0y′=sintcos3t,y(3π​)=0

Use Euler's method with n = 5 on the interval 0 ≤ t ≤ 12\frac { 1 } { 2 }21​ to approximate the solution f(t) to y′=3y,y(0)=1y ^ { \prime } = 3 y , y ( 0 ) = 1y′=3y,y(0)=1 Is the following the correct answer? =0;=1 =0.1;=1.3 =0.2;=1.69 3=0.3;=2.197 =0.4;=2.8561 5=0.5; y5 =3.71293

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Solve the differential equation: y' = $e ^ { y }$ sin t.

Solve the differential equation with the given initial condition. - $y ^ { \prime } = y ^ { 2 } , y ( 0 ) = 1$

A jug of milk at 50° is placed outdoors at a temperature of 100°. If after 5 minutes the temperature of the milk is 60°, write the equation giving the temperature of the milk as a function of time. Enter your answer exactly as: T = a $\mathrm { e } ^ { \mathrm { b } }$ + c

Suppose that y = f(t) satisfies the differential equation $y ^ { \prime }$ = $t ^ { 2 }$ + 3y, f(0) = 2. If Euler's method with $\mathrm { n } = 10$ is used to construct an approximation p(t) to f(t) for 0 ≤ t ≤ 1, find p(0.1).

Given the differential equation: $y ^ { \prime } = 3 t ^ { 2 } ( y - 7 )$ , is this the solution $y = 7 + \mathrm { ce } ^ { t ^ { 3 } } ?$

Given the differential equation with the given initial condition: $\frac { d y } { d t } = 3 t ^ { 2 } + \sin t ; y ( 0 ) = 2$ is this the solution $y = t ^ { 3 } - \cos t + 1 ?$

Suppose the relationship between the price p, of a product and the weekly sales, s, of the product is given by the differential equation $\frac { d p } { d s } = - \frac { 1 } { 10 } ( s + 3 )$ Then

Given the differential equation with the given initial condition: $y ^ { \prime } = \sqrt { \frac { t + 1 } { y } } ; y ( 0 ) = 4$ is this the solution $y = \left( ( t + 1 ) ^ { 3 / 2 } + 4 \right) ^ { 2 / 3 } \text { ? }$

Find the constant solutions to the differential equation: $y ^ { \prime } = y ^ { 2 } e ^ { t } - 2 y e ^ { t }$ . Enter just one integer or two separated by a comma (no label).

Solve the differential equation with the given initial condition. - $y ^ { \prime } = \sin t \cos ^ { 3 } t , y \left( \frac { \pi } { 3 } \right) = 0$

Use Euler's method with n = 5 on the interval 0 ≤ t ≤ $\frac { 1 } { 2 }$ to approximate the solution f(t) to $y ^ { \prime } = 3 y , y ( 0 ) = 1$ Is the following the correct answer? =0;=1 =0.1;=1.3 =0.2;=1.69 3=0.3;=2.197 =0.4;=2.8561 5=0.5; y5 =3.71293